Why express the total area of the rectangles as the difference of two harmonic series?

Conditions

• The total area is initially expressed as a sum of reciprocals from $\frac{1}{n+1}$ to $\frac{1}{2n}$.
• The asymptotic formula $\sum_{i=1}^m \frac{1}{i} = \ln m + \gamma + \sigma_m$ is available.

Reasoning, step by step

1. Simplify the general term of the rectangle sum to $\frac{1}{n+k}$.
2. Rewrite the sum as $\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n}$.
3. Add and subtract the first $n$ terms of the harmonic series to form the difference: $(1 + \frac{1}{2} + \dots + \frac{1}{2n}) - (1 + \frac{1}{2} + \dots + \frac{1}{n})$.
4. Apply the asymptotic expansion to both partial sums to prepare for taking the limit.

Example

The video states: 'Through this identity transformation, the original summation is cleverly rewritten as the difference of two partial sums of the harmonic series... This successfully converts the discrete summation problem into a continuous expression involving logarithmic functions and infinitesimal quantities—paving the way for ultimately evaluating the area's limiting value.'

Common misconceptions

• Thinking that the difference of harmonic series is just an algebraic trick without a deeper purpose.
• Forgetting that the asymptotic formula requires the sum to start from 1, hence the need to add and subtract the first n terms.

Watch the explanation

Explore next

• What is the asymptotic expansion formula for the harmonic series?
• How does the Euler-Mascheroni constant cancel out in the limit?